THE MELLIN CENTRAL PROJECTION TRANSFORM

2017 ◽  
Vol 58 (3-4) ◽  
pp. 256-264
Author(s):  
JIANWEI YANG ◽  
LIANG ZHANG ◽  
ZHENGDA LU
Keyword(s):  
Radial Direction ◽  
Mellin Transform ◽  
Closed Curve ◽  
Chinese Characters ◽  
Central Projection ◽  
Affine Invariant ◽  
Closed Curves ◽  
Affine Invariants ◽  
Invariant Features ◽  
Special Case

The central projection transform can be employed to extract invariant features by combining contour-based and region-based methods. However, the central projection transform only considers the accumulation of the pixels along the radial direction. Consequently, information along the radial direction is inevitably lost. In this paper, we propose the Mellin central projection transform to extract affine invariant features. The radial factor introduced by the Mellin transform, makes up for the loss of information along the radial direction by the central projection transform. The Mellin central projection transform can convert any object into a closed curve as a central projection transform, so the central projection transform is only a special case of the Mellin central projection transform. We prove that closed curves extracted from the original image and the affine transformed image by the Mellin central projection transform satisfy the same affine transform relationship. A method is provided for the extraction of affine invariants by employing the area of closed curves derived by the Mellin central projection transform. Experiments have been conducted on some printed Chinese characters and the results establish the invariance and robustness of the extracted features.

scholarly journals Extraction of Affine Invariant Features Using Fractal

2013 ◽  
Vol 2013 ◽  
pp. 1-8 ◽  
Author(s):  
Jianwei Yang ◽  
Guosheng Cheng ◽  
Ming Li
Keyword(s):  
Pattern Recognition ◽  
Feature Vector ◽  
Fractal Dimensions ◽  
Input Pattern ◽  
Experimental Results ◽  
Central Projection ◽  
Affine Invariant ◽  
Invariant Features ◽  
General Contour ◽  
New Feature

An approach based on fractal is presented for extracting affine invariant features. Central projection transformation is employed to reduce the dimensionality of the original input pattern, and general contour (GC) of the pattern is derived. Affine invariant features cannot be extracted from GC directly due to shearing. To address this problem, a group of curves (which are called shift curves) are constructed from the obtained GC. Fractal dimensions of these curves can readily be computed and constitute a new feature vector for the original pattern. The derived feature vector is used in question for pattern recognition. Several experiments have been conducted to evaluate the performance of the proposed method. Experimental results show that the proposed method can be used for object classification.

Radon–Fourier descriptor for invariant pattern recognition

2019 ◽  
Vol 17 (02) ◽  
pp. 1940004 ◽  
Author(s):  
Jianwei Yang ◽  
Liang Zhang ◽  
Peiyao Li
Keyword(s):  
Pattern Recognition ◽  
Radon Transform ◽  
Mellin Transform ◽  
Closed Curve ◽  
Fourier Descriptor ◽  
Binary Images ◽  
Invariant Features ◽  
Special Cases ◽  
Robust To Noise

Radon transform is not only robust to noise, but also independent on the calculation of pattern centroid. In this paper, Radon–Mellin transform (RMT), which is a combination of Radon transform and Mellin transform, is proposed to extract invariant features. RMT converts any object into a closed curve. Radon–Fourier descriptor (RFD) is derived by applying Fourier descriptor to the obtained closed curve. The obtained RFD is invariant to scaling and rotation. (Generic) R-transform and some other Radon-based methods can be viewed as special cases of the proposed method. Experiments are conducted on some binary images and gray images.

Quasi Fourier-Mellin Transform for Affine Invariant Features

2020 ◽  
Vol 29 ◽  
pp. 4114-4129
Author(s):  
Jianwei Yang ◽  
Zhengda Lu ◽  
Yuan Yan Tang ◽  
Zhou Yuan ◽  
Yunjie Chen
Keyword(s):  
Mellin Transform ◽  
Affine Invariant ◽  
Invariant Features ◽  
Fourier Mellin Transform

scholarly journals Cutting Affine Moment Invariants

2012 ◽  
Vol 2012 ◽  
pp. 1-12 ◽  
Author(s):  
Jianwei Yang ◽  
Ming Li ◽  
Zirun Chen ◽  
Yunjie Chen
Keyword(s):  
Image Processing ◽  
Closed Curve ◽  
Experimental Results ◽  
Moment Invariants ◽  
Affine Invariant ◽  
Original Image ◽  
Invariant Features ◽  
General Contour ◽  
Classification Tasks ◽  
Affine Moment Invariants

The extraction of affine invariant features plays an important role in many fields of image processing. In this paper, the original image is transformed into new images to extract more affine invariant features. To construct new images, the original image is cut in two areas by a closed curve, which is called general contour (GC). GC is obtained by performing projections along lines with different polar angles. New image is obtained by changing gray value of pixels in inside area. The traditional affine moment invariants (AMIs) method is applied to the new image. Consequently, cutting affine moment invariants (CAMIs) are derived. Several experiments have been conducted to evaluate the proposed method. Experimental results show that CAMIs can be used in object classification tasks.

RADIAL CENTROID CURVE FOR AFFINE INVARIANT FEATURES EXTRACTION

2012 ◽  
Vol 10 (04) ◽  
pp. 1250035 ◽  
Author(s):  
JIANWEI YANG ◽  
RUSHI LAN ◽  
YUAN YAN TANG ◽  
YUNJIE CHEN
Keyword(s):  
Computer Vision ◽  
Wavelet Transform ◽  
Line Segment ◽  
Invariant Function ◽  
Closed Curve ◽  
Affine Invariant ◽  
Stationary Wavelet Transform ◽  
Radial Line ◽  
Invariant Features ◽  
Wavelet Methods

The extraction of affine invariant features plays an important role in many fields of computer vision. Contour-based wavelet methods are unapplicable to objects with several separable components. In this paper, a method is proposed by converting the object into a closed curve, which is called radial centroid curve (RCC). Point on this curve is the centroid of radial line segment from centroid of the object. It is shown that the RCC derived from the affine transformed object is the same affine transformed version as that of the original object. An affine invariant function (AIF) is constructed by applying stationary wavelet transform (SWT) to the derived RCC. This scheme is applicable to objects with several separable components. Several experiments have been conducted to evaluate the performance of the proposed method.

On the Maximum Curvature of Closed Curves in Negatively Curved Manifolds

2015 ◽  
Vol 58 (4) ◽  
pp. 713-722
Author(s):  
Simon Brendle ◽  
Otis Chodosh
Keyword(s):  
Maximum Principle ◽  
Geodesic Curvature ◽  
Sharp Inequality ◽  
Closed Curve ◽  
Closed Curves ◽  
The Maximum Principle ◽  
Curved Manifolds ◽  
Maximum Curvature ◽  
Negatively Curved ◽  
Simply Connected

AbstractMotivated by Almgren’s work on the isoperimetric inequality, we prove a sharp inequality relating the length and maximum curvature of a closed curve in a complete, simply connected manifold of sectional curvature at most −1. Moreover, if equality holds, then the norm of the geodesic curvature is constant and the torsion vanishes. The proof involves an application of the maximum principle to a function defined on pairs of points.

scholarly journals Monotone homotopies and contracting discs on Riemannian surfaces

2018 ◽  
Vol 10 (02) ◽  
pp. 323-354 ◽  
Author(s):  
Gregory R. Chambers ◽  
Regina Rotman
Keyword(s):  
Finite Volume ◽  
Minimal Surfaces ◽  
Analogous Result ◽  
Simple Closed Curve ◽  
Closed Curve ◽  
Riemannian Surface ◽  
Closed Curves ◽  
Minimal Hypersurfaces ◽  
Riemannian Surfaces ◽  
Complete Manifolds

A monotone homotopy is a homotopy composed of simple closed curves which are also pairwise disjoint. In this paper, we prove a “gluing” theorem for monotone homotopies; we show that two monotone homotopies which have appropriate overlap can be replaced by a single monotone homotopy. The ideas used to prove this theorem are used in [G. R. Chambers and Y. Liokumovich, Existence of minimal hypersurfaces in complete manifolds of finite volume, arXiv:1609.04058] to prove an analogous result for cycles, which forms a critical step in their proof of the existence of minimal surfaces in complete non-compact manifolds of finite volume. We also show that, if monotone homotopies exist, then fixed point contractions through short curves exist. In particular, suppose that [Formula: see text] is a simple closed curve of a Riemannian surface, and that there exists a monotone contraction which covers a disc which [Formula: see text] bounds consisting of curves of length [Formula: see text]. If [Formula: see text] and [Formula: see text], then there exists a homotopy that contracts [Formula: see text] to [Formula: see text] over loops that are based at [Formula: see text] and have length bounded by [Formula: see text], where [Formula: see text] is the diameter of the surface. If the surface is a disc, and if [Formula: see text] is the boundary of this disc, then this bound can be improved to [Formula: see text].

Taming Wild Simple Closed Curves with Monotone Maps

1972 ◽  
Vol 24 (5) ◽  
pp. 768-788
Author(s):  
W. S. Boyd ◽  
A. H. Wright
Keyword(s):  
Simple Closed Curve ◽  
Closed Curve ◽  
Closed Curves ◽  
Monotone Map ◽  
Monotone Maps

Hempel [6, Theorem 2] proved that if S is a tame 2-sphere in E3 and f is a map of E3 onto itself such that f|S is a homeomorphism and f(E3 - S) = E3- f(S), then f(S) is tame. Boyd [4] has shown that the converse is false; in fact, if S is any 2-sphere in E3, then there is a monotone map f of E3 onto itself such that f |S is a homeomorphism, f(E3 — S) = E3 — f(S), and f(S) is tame.It is the purpose of this paper to prove that the corresponding converse for simple closed curves in E3 is also false. We show in Theorem 4 that if J is any simple closed curve in a closed orientable 3-manifold M3, then there is a monotone map f : M3 → S3 such that f |J is a homeomorphism, f(J) is tame and unknotted, and f(M3 - J) = S3 - f(J).In Theorem 1 of § 2, we construct a cube-with-handles neighbourhood of a simple closed curve in an orientable 3-manifold.

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